Question Video: Simplifying Expressions with Complex Numbers Mathematics • 12th Grade

Video Transcript

Simplify negative 12 minus four 𝑖 all over two 𝑖.

We have two ways in which we can divide a complex number by a purely imaginary number. Let’s think about both of those methods. The first of these methods involves rewriting the fraction. We essentially reverse the rules for adding and subtracting fractions to write negative 12 minus four 𝑖 over two 𝑖 as negative 12 over two 𝑖 minus four 𝑖 over two 𝑖. And now we see that if we look at the second fraction, we can simplify it quite nicely. 𝑖 divided by 𝑖 is one. Then, four divided by two is two, so this second fraction simplifies to simply negative two.

We can also do a little bit of simplifying on our first fraction. We divide both the numerator and the denominator by two. And so our expression becomes negative six over 𝑖 minus two. But we still have negative six over 𝑖 causing us some problems. And so we’re going to use the key facts that 𝑖 squared is equal to negative one. This means if we can multiply the denominator of our fraction by 𝑖, we will get a purely real number. We’ll get negative one. But of course, we have to do the same to the numerator. So we’re going to multiply both the numerator and the denominator of this fraction by 𝑖.

That gives us negative six 𝑖 over 𝑖 squared minus two. But we now know that 𝑖 squared is negative one. So it in fact gives us negative six 𝑖 over negative one minus two. We can then divide negative six 𝑖 by negative one, remembering that a negative divided by a negative gives a positive result. And we get simply six 𝑖 minus two or negative two plus six 𝑖. So, that’s method one. Let’s now consider method two.

In method two, we’re going to rewrite our purely imaginary denominator. We’re going to write it as zero plus two 𝑖 so that it essentially looks like the general form of a complex number. And then we recall that to divide by a complex number, we write it as a fraction and then multiply both the numerator and the denominator of the conjugate of the denominator. And to find the conjugate, we simply change the sign of the imaginary part. So, for a complex number of the form 𝑎 plus 𝑏𝑖, its conjugate 𝑧 bar is 𝑎 minus 𝑏𝑖. And this means the conjugate of zero plus two 𝑖 is zero minus two 𝑖.

Now that we’ve seen where this negative two 𝑖 comes from, we’re going to get rid of all the zeros. And so we’re simply going to multiply the numerator and denominator of our fraction by negative two 𝑖. Let’s begin with the numerator. We’re going to work out negative two 𝑖 times negative 12, which is 24𝑖. And then we’re going to work out negative two 𝑖 times negative four 𝑖, which is eight 𝑖 squared. Once again, though, we know that 𝑖 squared is negative one. So we get 24𝑖 plus eight times negative one, which is 24𝑖 minus eight.

Then we work out the value of the denominator. It’s two 𝑖 times negative two 𝑖. That’s negative four 𝑖 squared, which can, of course, be written as negative four times negative one, which is simply four. And so when we multiply the numerator and the denominator of our earlier fraction by negative two 𝑖, we get 24𝑖 minus eight all over four. And then we know we can divide both parts of our numerator by four. So we get six 𝑖 minus two, which once again is equal to negative two plus six 𝑖.

Of course, both methods are perfectly valid here. Either way, we see that we simplified negative 12 minus four 𝑖 over two 𝑖. And we get negative two plus six 𝑖.